Global element

Definition

A constant or global element of a category $𝐂$ is a morphism $1\to K$, where $1$ is the terminal object of $𝐂$. Constants are named so as they identify single "elements" of an object. E.g., a constant in $𝐒𝐞𝐭$ would choose a single element of a set.

More generally, a constant is a morphism $c:y\to z$ with the property that, for any two morphisms $f,g:x\to y$, $fc=gc$.[cite:@nlab:constant_morphism]

Motivation

Constants are a tool essential to the object-apathetic categorical wordsmith. For instance, one may wish to define the behaviour of a function $\neg :\text{Bool}\to \text{Bool}$ in categorical language. Since the categorist may not use objects, as in the traditional definition,

they can instead express $\text{True}$ and $\text{False}$ as distinct morphisms $\{\ast \}\to \text{Bool}$ and describe the behaviour of $\neg$ using composition, as in

or diagrammatically as in

(quiver)

References

• ProofWiki.org — Constant (Category Theory)

• [cite:@barr1990category]

• [cite:@riehl2017category]